OFFSET
0,3
COMMENTS
The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 60 and Q = -1. This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 26 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,62,0,-1).
FORMULA
G.f.: -(x^2-x-1) / ((x^2-8*x+1)*(x^2+8*x+1)). - Colin Barker, Dec 25 2013
a(n) = 62*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Dec 25 2013
From Peter Bala, May 26 2014: (Start)
The following remarks assume an offset of 1:
Let alpha = sqrt(15) + 4 and beta = sqrt(15) - 4 be the roots of the equation x^2 - sqrt(60)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = product {k = 1..floor((n-1)/2)} ( 60 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 60*a(2*n) + a(2*n - 1). (End)
MATHEMATICA
Denominator[Convergents[Sqrt[960], 30]] (* Vincenzo Librandi, Dec 25 2013 *)
PROG
(Magma) I:=[1, 1, 61, 62]; [n le 4 select I[n] else 62*Self(n-2)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Dec 25 2013
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
N. J. A. Sloane, Dec 11 1999
EXTENSIONS
More terms from Colin Barker, Dec 25 2013
STATUS
approved